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November  2016, 15(6): 2059-2074. doi: 10.3934/cpaa.2016027

On the Hardy-Littlewood-Sobolev type systems

Ze Cheng 1, , Genggeng Huang 2, and  Congming Li 3,

1. 

Department of Applied Mathematics, University of Colorado at Boulder, Colorado
2. 

Department of Mathematics, INS and MOE-LSC, Shanghai Jiao Tong University, Shanghai, China
3. 

Department of Applied Mathematics, University of Colorado at Boulder

Received  October 2015 Revised  June 2016 Published  September 2016

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.
Keywords: existence, uniqueness., non-existence, Hardy-Littlewood-Sobolev.
Mathematics Subject Classification: Primary: 35J9.
Citation: Ze Cheng, Genggeng Huang, Congming Li. On the Hardy-Littlewood-Sobolev type systems. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2059-2074. doi: 10.3934/cpaa.2016027
References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317.  doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

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W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

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W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().   Google Scholar

[8]

Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2.  doi: 10.1016/j.na.2014.10.019.  Google Scholar

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.   Google Scholar

[10]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[11]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[12]

C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

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E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

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J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).   Google Scholar

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E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).   Google Scholar

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E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

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E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.   Google Scholar

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S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.   Google Scholar

[19]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

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P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).   Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rat. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[23]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.   Google Scholar

[24]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.   Google Scholar

[25]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[26]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

show all references

References:
[1]

F. Arthur, X. Yan and M. Zhao, A Liouville-type theorem for higher order elliptic systems,, \emph{Disc. & Cont. Dynamics Sys.}, 34 (2014), 3317.  doi: 10.3934/dcds.2014.34.3317.  Google Scholar

[2]

H. Berestycki, P. L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semi-linear problems in $R^N$,, \emph{Indiana University Mathematics Journal}, 30 (1981), 141.  doi: 10.1512/iumj.1981.30.30012.  Google Scholar

[3]

L. Caffarelli, B. Gidas and J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth,, \emph{Comm. Pure Appl. Math.}, 142 (1989), 615.  doi: 10.1002/cpa.3160420304.  Google Scholar

[4]

W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, \emph{Duke Math. J.}, 63 (1991), 615.  doi: 10.1215/S0012-7094-91-06325-8.  Google Scholar

[5]

W. Chen, C. Li and B. Ou, Classification of solutions for a system of integral equations,, \emph{Commun. in Partial Differential Equations}, 30 (2005), 59.  doi: 10.1081/PDE-200044445.  Google Scholar

[6]

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, \emph{Comm. Pure Appl. Math.}, 59 (2006), 330.  doi: 10.1002/cpa.20116.  Google Scholar

[7]

Z. Cheng, G. Huang and C. Li, A Liouville theorem for subcritical Lane-Emden system,, \arXiv{1412.7275}., ().   Google Scholar

[8]

Z. Cheng and C. Li, Shooting method with sign-changing nonlinearity,, \emph{Nonlinear Analysis: Theory, 114 (2015), 2.  doi: 10.1016/j.na.2014.10.019.  Google Scholar

[9]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in $R^n$,, \emph{Math. Anal. and Applications, 7A (1981), 369.   Google Scholar

[10]

Y. Lei and C. Li, Sharp criteria of Liouville type for some nonlinear systems,, \emph{Discrete Contin. Dyn. Syst.}, 36 (2016), 3277.  doi: 10.3934/dcds.2016.36.3277.  Google Scholar

[11]

Y. Lei, C. Li and C. Ma, Asymptotic radial symmetry and growth estimates of positive solutions to the weighted HLS system,, \emph{Calc. Var. of Partial Differential Equations}, 45 (2012), 43.  doi: 10.1007/s00526-011-0450-7.  Google Scholar

[12]

C. Li and J. Villaver, Existence of positive solutions to semilinear elliptic systems with supercritical growth,, \emph{Comm. in Partial Differential Equation}, 41 (2016), 1029.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

[13]

E. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities,, \emph{Ann. of Math.}, 118 (1983), 349.  doi: 10.2307/2007032.  Google Scholar

[14]

J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems,, \emph{Journal of Partial Differential Equations}, 19 (2006).   Google Scholar

[15]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Quaderno Matematico}, (1982).   Google Scholar

[16]

E. Mitidieri, A Rellich type identity and applications: Identity and applications,, \emph{Communications in partial differential equations}, 18 (1993), 125.  doi: 10.1080/03605309308820923.  Google Scholar

[17]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $R^N$,, \emph{Differ. Integral Equations}, 9 (1996), 465.   Google Scholar

[18]

S. Pohozaev, Eigenfunctions of the equation $\Delta u + \lambda f(u)=0$,, \emph{Soviet Math. Doklady}, 6 (1965), 1408.   Google Scholar

[19]

P. Polacik, P. Quittner and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,, \emph{Duke Math. J.}, 139 (2007), 555.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[20]

P. Pucci and J. Serrin, A general variational identity,, \emph{Indiana Univ. J. Math.}, 35 (1986), 681.  doi: 10.1512/iumj.1986.35.35036.  Google Scholar

[21]

Pavol Quittner and Philippe Souplet, Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States,, Springer, (2007).   Google Scholar

[22]

J. Serrin, A symmetry problem in potential theory,, \emph{Arch. Rat. Mech. Anal.}, 43 (1971), 304.   Google Scholar

[23]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane-Emden systems,, \emph{Differ. Integral Equations}, 9 (1996), 635.   Google Scholar

[24]

J. Serrin and H. Zou, Existence of positive solutions of the Lane-Emden system,, \emph{Atti Semi. Mat. Fis. Univ. Modena}, 46 (1998), 369.   Google Scholar

[25]

P. Souplet, The proof of the Lane-Emden conjecture in four space dimensions,, \emph{Advances in Mathematics}, 221 (2009), 1409.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[26]

J. Wei and X. Xu, Classification of solutions of higher order conformally invariant equations,, \emph{Math. Ann.}, (1999), 207.  doi: 10.1007/s002080050258.  Google Scholar

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